Similar Triangles and Slope CC8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non‐vertical line in the coordinateplane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Similar Triangles• Similar Triangles are triangles who have the same shape, but not necessarily the same size. The corresponding angles of similar triangles are congruent and their corresponding sides are in PROPORTION. The similar triangles increase or decrease at a constant rate.
How do I know if two triangles are similar? .If two triangles are similar, the cross products of theircorresponding sides are equal. 5 3 10 6 5 units 10 3 5 6 10 units 30 30Since the cross products of the 3 unitscorresponding sides are equal, the 6 unitstriangles are similar.
Rates of Proportionality in a Triangle? Make a rate ofthe legs in each of these right triangles and compare the results. When making your rate, compare the vertical leg (rise) to the horizontal leg (run).
What did you notice? 4The red triangle has a rate of 4 to 8 or 8 4 8 5The blue triangle has a rate of 5 to 10 or 10 5 10 3The green triangle has a rate of 3 to 6 or 6 3 6
How many triangles do you see?Find the ratio of verticalto horizontal leg of eachtriangle. Then simplify toa fraction. The simplifiedfraction should be theSLOPE of the red line. 3 7 9 3 7 9The SLOPE of the red lineis 1 because all of theslope ratios simplify to 1.
The rate of each triangle can be simplified to ½ ! What do you notice about these triangles and their hypotenuse in the illustration below?
Positive slope Negative slopeRises from left Falls from left toto right right rise 2 run 3 Zero slope Undefined slope Horizontal line Vertical linerise 0 0run 5 rise 5 Undefined run 0
Draw triangles to find the slope of the line. The slope of the red line is negative since the triangles are moving down. For the smaller triangle, the vertical change is 2 and the horizontal change is 3. For the larger triangle, the vertical change is 4 and the horizontal change is 6. 4 The slope for the red line must be or 2 . 6 3